# The Stacks Project

## Tag 02PI

Lemma 41.3.3. Let $R$ be a local ring.

1. If $(M, N, \varphi, \psi)$ is a $2$-periodic complex such that $M$, $N$ have finite length. Then $e_R(M, N, \varphi, \psi) = \text{length}_R(M) - \text{length}_R(N)$.
2. If $(M, \varphi, \psi)$ is a $(2, 1)$-periodic complex such that $M$ has finite length. Then $e_R(M, \varphi, \psi) = 0$.
3. Suppose that we have a short exact sequence of $2$-periodic complexes $$0 \to (M_1, N_1, \varphi_1, \psi_1) \to (M_2, N_2, \varphi_2, \psi_2) \to (M_3, N_3, \varphi_3, \psi_3) \to 0$$ If two out of three have cohomology modules of finite length so does the third and we have $$e_R(M_2, N_2, \varphi_2, \psi_2) = e_R(M_1, N_1, \varphi_1, \psi_1) + e_R(M_3, N_3, \varphi_3, \psi_3).$$

Proof. Proof of (3). Abbreviate $A = (M_1, N_1, \varphi_1, \psi_1)$, $B = (M_2, N_2, \varphi_2, \psi_2)$ and $C = (M_3, N_3, \varphi_3, \psi_3)$. We have a long exact cohomology sequence $$\ldots \to H^1(C) \to H^0(A) \to H^0(B) \to H^0(C) \to H^1(A) \to H^1(B) \to H^1(C) \to \ldots$$ This gives a finite exact sequence $$0 \to I \to H^0(A) \to H^0(B) \to H^0(C) \to H^1(A) \to H^1(B) \to K \to 0$$ with $0 \to K \to H^1(C) \to I \to 0$ a filtration. By additivity of the length function (Algebra, Lemma 10.51.3) we see the result. The proofs of (1) and (2) are omitted. $\square$

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\begin{lemma}
\label{lemma-periodic-length}
Let $R$ be a local ring.
\begin{enumerate}
\item If $(M, N, \varphi, \psi)$ is a $2$-periodic complex
such that $M$, $N$ have finite length. Then
$e_R(M, N, \varphi, \psi) = \text{length}_R(M) - \text{length}_R(N)$.
\item If $(M, \varphi, \psi)$ is a $(2, 1)$-periodic complex
such that $M$ has finite length. Then
$e_R(M, \varphi, \psi) = 0$.
\item Suppose that we have a short exact sequence of
$2$-periodic complexes
$$0 \to (M_1, N_1, \varphi_1, \psi_1) \to (M_2, N_2, \varphi_2, \psi_2) \to (M_3, N_3, \varphi_3, \psi_3) \to 0$$
If two out of three have cohomology modules of finite length so does
the third and we have
$$e_R(M_2, N_2, \varphi_2, \psi_2) = e_R(M_1, N_1, \varphi_1, \psi_1) + e_R(M_3, N_3, \varphi_3, \psi_3).$$
\end{enumerate}
\end{lemma}

\begin{proof}
Proof of (3). Abbreviate $A = (M_1, N_1, \varphi_1, \psi_1)$,
$B = (M_2, N_2, \varphi_2, \psi_2)$ and $C = (M_3, N_3, \varphi_3, \psi_3)$.
We have a long exact cohomology sequence
$$\ldots \to H^1(C) \to H^0(A) \to H^0(B) \to H^0(C) \to H^1(A) \to H^1(B) \to H^1(C) \to \ldots$$
This gives a finite exact sequence
$$0 \to I \to H^0(A) \to H^0(B) \to H^0(C) \to H^1(A) \to H^1(B) \to K \to 0$$
with $0 \to K \to H^1(C) \to I \to 0$ a filtration. By additivity of
the length function (Algebra, Lemma \ref{algebra-lemma-length-additive})
we see the result.
The proofs of (1) and (2) are omitted.
\end{proof}

Comment #2608 by Ko Aoki on June 24, 2017 a 6:02 am UTC

Typo in the statement of (3): "$(2,1)$-periodic complexes" should be replaced by "$2$-periodic complexes."

Comment #2631 by Johan (site) on July 7, 2017 a 12:48 pm UTC

Thanks, fixed here.

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